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arXiv:2501.00327 · Physical Review A

Investigating pure state uniqueness in tomography via optimization

J. Wu, Zheng An, Chao Zhang, Xuanran Zhu, Shilin Huang, Bei Zeng

Published Sep 29, 2025• 1 citation
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Abstract

Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring $\mathcal{O}({d}^{2})$ measurements for a general $d$-dimensional state. This complexity can be substantially reduced to $\mathcal{O}(d)$ in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among aall states (UDA). The UDP determination inherently involves nonconvexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the mugmented Lagrangian ethod (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parametrizing quantum states and employing ALM to handle the constrained nonconvex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems, generalized Greenberger-Horne-Zeilinger states and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (a) UDA, (b) UDP but not UDA, and (c) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.

DOI
10.1103/p7kb-k52j

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Quantum state tomography (QST) is crucial for understanding and characterizing quantum systems through measurement data. Traditional QST methods face scalability challenges, requiring O(d2)\mathcal{O}({d}^{2})O(d2) measurements for a general ddd-dimensional state. This complexity can be substantially reduced to O(d)\mathcal{O}(d)O(d) in pure state tomography, indicating that full measurements are unnecessary for pure states. In this paper, we investigate the conditions under which a given pure state can be uniquely determined by a subset of full measurements, focusing on the concepts of uniquely determined among pure states (UDP) and uniquely determined among aall states (UDA). The UDP determination inherently involves nonconvexity challenges, while the UDA determination, though convex, becomes computationally intensive for high-dimensional systems. To address these issues, we develop a unified framework based on the mugmented Lagrangian ethod (ALM). Specifically, our theorem on the existence of low-rank solutions in QST allows us to reformulate the UDA problem with low-rank constraints, thereby reducing the number of variables involved. Our approach entails parametrizing quantum states and employing ALM to handle the constrained nonconvex optimization tasks associated with UDP and low-rank UDA determinations. Numerical experiments conducted on qutrit systems, generalized Greenberger-Horne-Zeilinger states and four-qubit symmetric states not only validate theoretical findings but also reveal the complete distribution of quantum states across three uniqueness categories: (a) UDA, (b) UDP but not UDA, and (c) neither UDP nor UDA. This work provides a practical approach for determining state uniqueness, advancing our understanding of quantum state reconstruction.
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Full Pauli-basis tomography of an arbitrary nnn-qubit state uses the expectation values of all nonidentity Pauli observables. For multiqubit systems, prior knowledge that the state is pure allows some of these observables to be omitted while retaining uniqueness. We study two resulting notions of uniqueness: whether every pure state is uniquely determined among pure states (UDP), and whether every pure state is uniquely determined among all states, including mixed states (UDA). We derive quartic uniqueness certificates from the fourth elementary symmetric polynomial of operators in the real span of the omitted Pauli observables. An exact quartic identity shows that, for nnn-qubit systems with n≥3n\geq 3n≥3, every pure state is UDA whenever the omitted Pauli observables contain no four distinct operators that commute pairwise and multiply to ±I\pm I±I. For three qubits, this condition is also necessary for UDP, yielding an exact characterization of both UDP and UDA. We further prove that UDP and UDA coincide for any number of qubits whenever the omitted Pauli observables commute pairwise. These results provide directly checkable structural criteria for certifying pure-state tomography from restricted Pauli measurements.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
Quantum state tomography (QST) aims at estimating a quantum state from averaged quantum measurements made on copies of that state. Most quantum algorithms rely on QST at some point and it is a well-explored topic in the literature, mostly for mixed states. In this paper we focus on the QST of a pure quantum state using parallel unentangled measurements. Pure states are a small but useful subset of all quantum states, and their tomography requires fewer measurements and is essentially a phase recovery problem. Parallel unentangled measurements are easy to implement in practice because they allow the user to measure each qubit individually, e.g., using one-qubit Pauli measurements. We propose two sets of quantum measurements that one can make on a pure state as well as the algorithms that use the measurement outcomes in order to identify the state. We also discuss how those estimates can be fine tuned by finding the state that maximizes the likelihood of the measurements with different variants of the likelihood. The performances of the proposed three types of QST methods are validated by means of detailed numerical tests, including for mixed states that are close to being pure.
With nowadays steadily growing quantum processors, it is required to develop new quantum tomography tools that are tailored for high-dimensional systems. In this work, we describe such a computational tool, based on recent ideas from non-convex optimization. The algorithm excels in the compressed-sensing-like setting, where only a few data points are measured from a low-rank or highly-pure quantum state of a high-dimensional system. We show that the algorithm can practically be used in quantum tomography problems that are beyond the reach of convex solvers, and, moreover, is faster than other state-of-the-art non-convex approaches. Crucially, we prove that, despite being a non-convex program, under mild conditions, the algorithm is guaranteed to converge to the global minimum of the problem; thus, it constitutes a provable quantum state tomography protocol.
Quantum state tomography (QST) is a crucial technique for characterizing the state of a quantum system, often employed in quantum metrology, quantum communication, and fundamental quantum information science. However, QST inherently suffers from significant measurement redundancy and uncertainty, demanding a large number of measurements to achieve a desired level of accuracy. This work proposes a novel framework for QST leveraging concepts from algorithmic information theory, specifically Kolmogorov complexity, to rigorously quantify this uncertainty and redundancy. We introduce a tailored complexity measure for quantum states that allows for the design of measurement strategies optimized to minimize the number of required measurements while maintaining a specified accuracy. Our approach shifts the focus from purely statistical estimation to a fundamental information-theoretic analysis, offering a deeper understanding of the reconstruction problem and potentially leading to more efficient and robust QST protocols. The core claim is that applying Kolmogorov complexity provides a quantifiable metric for the inherent uncertainty within a quantum state, directly informing the development of improved estimation techniques. The mechanism involves defining and utilizing this complexity measure to guide the selection of measurement angles, ultimately reducing redundancy and improving the precision of state reconstruction.